In laser resonators, higher order modes (i.e. TEM01, etc) accumulate phase faster than the fundamental TEM00 mode. This extra phase is called Gouy phase. What is an intuitive explanation of this effect?

Gouy predicted and then experimentally verified the existence of this effect long before the existence of lasers. How did he do it, and what motivated him to think about it?

2 Answers
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Several sources link to this paper: S. Feng, H. G. Winful, Physical origin of the Gouy phase shift, Optics Letters, 26, 485 (2001), which tries to give an intuitive explanation of the Gouy phase. Briefly, the point is that convergent waves going through the focus have finite spatial extent in the transverse plane. The uncertainty relation induces then some distribution over the transverse and consequently longitudinal wave vectors. It is claimed that the net effect of this distribution over wave vectors is an overall phase shift, which is larger for higher modes. However to see that one really needs to look into the formulas.

A naive explanation says that a beam after its focal point is inverted, not only in the sense of its spatial distribution but also in the sense of the direction of the electrical field vector (minus sign = adding $\pi$ to the phase). It's perfectly compatible with the fact why even beam profiles change the phase by $\pi$ and odd do not.

However, this explanations says nothing about behaviour of the phase near the focal point.